In math, sometimes the most common things are the hardest to find.
Quanta Magazine said:The first time I heard a mathematician use the phrase, I was sure he’d misspoken. We were on the phone, talking about the search for shapes with certain properties, and he said, “It’s like looking for hay in a haystack.”
“Don’t you mean a needle?” I almost interjected. Then he said it again.
In mathematics, it turns out, conventional modes of thought sometimes get turned on their head. The mathematician I was speaking with, Dave Jensen of the University of Kentucky, really did mean “hay in a haystack.” By it, he was expressing a strange fact about mathematical research: Sometimes the most common things are the hardest to find.
“In many areas of mathematics you’re looking for examples of something, and examples are really abundant, but somehow any time you try to write down an example, you get it wrong,” said Jensen.
The hay-in-a-haystack phenomenon is at work in one of the first objects that kids encounter in mathematics: the number line. Points on the number line include the positive and negative integers (such as 2 and –29), rational numbers (ratios of integers like 32 and 1137) and all irrational numbers — those numbers, like pi or 2‾√, that can’t be expressed as a ratio.
Irrational numbers occupy the vast, vast majority of space on a number line — so vast, in fact, that if you were to pick a number on the number line at random, there is literally a 100 percent chance that it will be irrational.
Yet despite their overwhelming presence, we almost never encounter irrational numbers in our daily lives. Instead we count with whole numbers and follow recipes with fractions. The numbers we know best are the extremely rare numbers, the special numbers — the needles in the haystack.
The hay is hard to find precisely because it’s so unexceptional. Rational numbers have the distinctive property that it’s possible to write them down. This calls them to our attention. Irrational numbers have an infinite decimal expansion. You couldn’t write one down even with an endless amount of time. That these numbers lack the exceptional property of “write-down-able-ness” is what makes them nearly invisible to our way of seeing.
“We’re looking with a magnet, and you’re not going to find hay with magnet; you’re only going to find needles,” said Dhruv Ranganathan, a mathematician who is in the midst of a move from the Massachusetts Institute of Technology to the University of Cambridge.
...